Gelfand-Levitan-Krein method in one-dimensional elasticity inverse problem

In this article we propose the numerical solution of the one dimensional inverse coefficient problem for seismic equation. We use a dynamical version of Gelfand-Levitan-Krein approach for reducing a nonlinear inverse problem for recovering the shear wave’s velocity and the density of the medium to two sequences of the linear integral equations. We propose numerical algorithm for solving these equations based on a fast inversion of a Toeplitz matrix. The proposed numerical methods base on the structure of the problem and therefore improve the efficiency of the algorithms, compared with standard approaches. We present numerical results for solving considered integral equations.

Simultaneous identification and reconstruction of the space-dependent reaction coefficient and source term

The inverse problem of simultaneously determining, i.e., identifying and reconstructing, the space-dependent reaction coefficient and source term component from time-integral temperature measurements is investigated. This corresponds to thermal applications in which the heat is generated from a source depending linearly on the temperature, but with unknown space-dependent coefficients. For the resulting nonlinear inverse problem, we first prove the existence of solution based on the Schauder fixed point theorem. Then, under certain additional conditions, the solution is also proved to be unique. For the numerical reconstruction of solution, the problem is reformulated as a least-squares minimisation whose Fréchet gradients with respect to the two unknowns are derived in terms of the solution of an adjoint problem. The conjugate gradient method (CGM) to calculate the numerical solution is developed, and its convergence is proved from the Lipschitz continuity of these gradients. Three numerical examples for one- and two-dimensional inverse problems are illustrated to reveal the accuracy and stability of the solutions applying the CGM regularised by the discrepancy principle when noisy data are inverted.

Improved Error Reduction and Hybrid Input Output Algorithms for Phase Retrieval by including a Sparse Dictionary Learning-Based Inpainting Method

The phase retrieval (PR), reconstructing an object from its Fourier magnitudes, is equivalent to a nonlinear inverse problem. In this paper, we proposed a two-step algorithm that traditional ER/HIO iteration plays as the coarse feature reconstruction, whereas the KSVD-based inpainting technique deals with the fine feature set accordingly. Since the KSVD allows the content of oversampled dictionary with sparse representation to adaptively fit a given set of object examples, as long as the ER/HIO algorithms provide decent object estimation at early stage, the pixels violating the object constraint can be restored with superior image quality. The numerical analyses demonstrated the effectiveness of ER + KSVD and HIO + KSVD through multiple independent initial Fourier phases. With its versatility and simplicity, the proposed method can be generalized to be implemented with more PR state-of-the-arts.

Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system

The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.